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October  2018, 23(8): 3047-3070. doi: 10.3934/dcdsb.2017196

On the scale dynamics of the tropical cyclone intensity

Chanh Kieu 1,, and  Quan Wang 1,2,

1. 

Department of Earth and Atmospheric Sciences, Indiana University, Bloomington, IN 47405, USA
2. 

Department of Mathematics, Sichuan University, Sichuan Sheng, China

* Corresponding author: ckieu@indiana.edu

Received  March 2017 Published  July 2017

Fund Project: This research was supported by the NOAA HFIP funding (Award NA16NWS4680026), the ONR funding (Grant N000141410143), and the Vice Provost for Research through the Indiana University Faculty Research Support Program.

This study examines the dynamics of tropical cyclone (TC) development in a TC scale framework. It is shown that this TC-scale dynamics contains the maximum potential intensity (MPI) limit as an asymptotically stable point for which the Coriolis force and the tropospheric stratification are two key parameters responsible for the bifurcation of TC development. In particular, it is found that the Coriolis force breaks the symmetry of the TC development and results in a larger basin of attraction toward the cyclonic (anticyclonic) stable point in the Northern (Southern) Hemisphere. Despite the sensitive dependence of intensity bifurcation on these two parameters, the structurally stable property of the MPI critical point is maintained for a wide range of parameters.

Keywords: Stability, tropical cyclones, intensity equilibrium, scale analysis, hurricane dynamics.
Mathematics Subject Classification: Primary: 76U05, 76E09; Secondary: 58F15, 58F17.
Citation: Chanh Kieu, Quan Wang. On the scale dynamics of the tropical cyclone intensity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3047-3070. doi: 10.3934/dcdsb.2017196
References:
[1]

B. R. Brown and G. J. Hakim, Variability and predictability of a three-dimensional hurricane in statistical equilibrium, J. Atmos. Sci., 70 (2013), 1806-1820.  doi: 10.1175/JAS-D-12-0112.1.  Google Scholar

[2]

G. H. Bryan and R. Rotunno, The maximum intensity of tropical cyclones in axisymmetric numerical model simulations, Mon. Wea. Rev, 137 (2009), 1770-1789.  doi: 10.1175/2008MWR2709.1.  Google Scholar

[3]

J. G. Charney and A. Eliassen, On the growth of the hurricane depression, J. Atmos. Sci, 21 (1964), 68-75.  doi: 10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2.  Google Scholar

[4]

K. A. Emanuel, A statistical analysis of tropical cyclone intensity, Monthly Weather Review, 128 (2000), 1139-1152.  doi: 10.1175/1520-0493(2000)128<1139:ASAOTC>2.0.CO;2.  Google Scholar

[5]

K. A. Emanuel, An air-sea interaction theory for tropical cyclones. part i: Steady-state maintenance, J. Atmos. Sci, 43 (1986), 585-605.  doi: 10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2.  Google Scholar

[6]

K. A. Emanuel, The Maximum Intensity of Hurricanes, J. Atmos. Sci, 45 (1986), 1143-1155.  doi: 10.1175/1520-0469(1988)045<1143:TMIOH>2.0.CO;2.  Google Scholar

[7]

M. Ferrara,, F. Groff, Z. Moon, K. Keshavamurthy, S. M. Robeson and C. Kieu, Large-scale control of the lower stratosphere on variability of tropical cyclone intensity, Geophys. Res. Lett., 44 (2017), 4313-4323.   Google Scholar

[8]

G. J. Hakim, The mean state of axisymmetric hurricanes in statistical equilibrium, J. Atmos. Sci, 68 (2011), 1364-1376.  doi: 10.1175/2010JAS3644.1.  Google Scholar

[9]

G. J. Hakim, The variability and predictability of axisymmetric hurricanes in statistical equilibrium, J. Atmos. Sci, 70 (2013), 993-1005.  doi: 10.1175/JAS-D-12-0188.1.  Google Scholar

[10]

K. A. Hill and G. M. Lackmann, The impact of future climate change on TC intensity and structure: A downscaling approach, Journal of Climate, 24 (2011), 4644-4661.  doi: 10.1175/2011JCLI3761.1.  Google Scholar

[11]

C. Kieu, Hurricane maximum potential intensity equilibrium, Q.J.R. Meteorol. Soc., 141 (2015), 2471-2480.  doi: 10.1002/qj.2556.  Google Scholar

[12]

C. Kieu and Z. Moon, Hurricane intensity predictability, Bull. Amer. Meteo. Soc., 97 (2016), 1847-1857.  doi: 10.1175/BAMS-D-15-00168.1.  Google Scholar

[13]

C. Kieu, H. Chen and D. L. Zhang, An examination of the pressure-wind relationship for intense tropical cyclones, Wea. and Forecasting, 25 (2010), 895-907.   Google Scholar

[14]

Y. Liu, D.-L. Zhang and M. K. Yau, A Multiscale Numerical Study of Hurricane Andrew Part Ⅱ: Kinematics and Inner-Core Structures, J. Atmos. Sci, 127 (1999), 2597-2616.   Google Scholar

[15]

T. Ma, Topology of Manifolds Science Press, Beijing. , 2007. Google Scholar

[16]

J. W. Milnor, Topology from the Differentiable Viewpoint Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va. 1965.  Google Scholar

[17]

Y. Ogura and N. A. Phillips, Scale analysis of deep and shallow convection in the atmophere, J. Atmos. Sci, 19 (1962), 173-179.   Google Scholar

[18]

R. Rotunno and K. A. Emanuel, An airsea interaction theory for tropical cyclones. part ii: Evolutionary study using a nonhydrostatic axisymmetric numerical model, J. Atmos. Sci, 44 (1987), 542-561.   Google Scholar

[19]

W. Shen, R. E. Tuleya and I. Ginis, A sensitivity study of the thermodynamic environment on GFDL model hurricane intensity: Implications for global warming, J. Climate, 13 (2000), 109-121.   Google Scholar

[20]

R. K. Smith, G. Kilroy and M. T. Montgomery, Why do model tropical cyclones intensify more rapidly at low latitudes, Journal of the Atmospheric Sciences, 72 (2015), 1783-1840.  doi: 10.1175/JAS-D-14-0044.1.  Google Scholar

[21]

R. Wilhelmson and Y. Ogura, The pressure perturbation and the numerical modeling of a cloud, Journal of the Atmospheric Sciences, 29 (1972), 1295-1307.  doi: 10.1175/1520-0469(1972)029<1295:TPPATN>2.0.CO;2.  Google Scholar

[22]

H. E. Willoughby, Forced secondary circulations in hurricanes, J. Geophys. Res, 84 (1979), 3173-3183.  doi: 10.1029/JC084iC06p03173.  Google Scholar

[23]

H. E. Willoughby, Gradient Balance in Tropical Cyclones, J. Atmos. Sci., 47 (1990), 265-274.  doi: 10.1175/1520-0469(1990)047<0265:GBITC>2.0.CO;2.  Google Scholar

show all references

References:
[1]

B. R. Brown and G. J. Hakim, Variability and predictability of a three-dimensional hurricane in statistical equilibrium, J. Atmos. Sci., 70 (2013), 1806-1820.  doi: 10.1175/JAS-D-12-0112.1.  Google Scholar

[2]

G. H. Bryan and R. Rotunno, The maximum intensity of tropical cyclones in axisymmetric numerical model simulations, Mon. Wea. Rev, 137 (2009), 1770-1789.  doi: 10.1175/2008MWR2709.1.  Google Scholar

[3]

J. G. Charney and A. Eliassen, On the growth of the hurricane depression, J. Atmos. Sci, 21 (1964), 68-75.  doi: 10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2.  Google Scholar

[4]

K. A. Emanuel, A statistical analysis of tropical cyclone intensity, Monthly Weather Review, 128 (2000), 1139-1152.  doi: 10.1175/1520-0493(2000)128<1139:ASAOTC>2.0.CO;2.  Google Scholar

[5]

K. A. Emanuel, An air-sea interaction theory for tropical cyclones. part i: Steady-state maintenance, J. Atmos. Sci, 43 (1986), 585-605.  doi: 10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2.  Google Scholar

[6]

K. A. Emanuel, The Maximum Intensity of Hurricanes, J. Atmos. Sci, 45 (1986), 1143-1155.  doi: 10.1175/1520-0469(1988)045<1143:TMIOH>2.0.CO;2.  Google Scholar

[7]

M. Ferrara,, F. Groff, Z. Moon, K. Keshavamurthy, S. M. Robeson and C. Kieu, Large-scale control of the lower stratosphere on variability of tropical cyclone intensity, Geophys. Res. Lett., 44 (2017), 4313-4323.   Google Scholar

[8]

G. J. Hakim, The mean state of axisymmetric hurricanes in statistical equilibrium, J. Atmos. Sci, 68 (2011), 1364-1376.  doi: 10.1175/2010JAS3644.1.  Google Scholar

[9]

G. J. Hakim, The variability and predictability of axisymmetric hurricanes in statistical equilibrium, J. Atmos. Sci, 70 (2013), 993-1005.  doi: 10.1175/JAS-D-12-0188.1.  Google Scholar

[10]

K. A. Hill and G. M. Lackmann, The impact of future climate change on TC intensity and structure: A downscaling approach, Journal of Climate, 24 (2011), 4644-4661.  doi: 10.1175/2011JCLI3761.1.  Google Scholar

[11]

C. Kieu, Hurricane maximum potential intensity equilibrium, Q.J.R. Meteorol. Soc., 141 (2015), 2471-2480.  doi: 10.1002/qj.2556.  Google Scholar

[12]

C. Kieu and Z. Moon, Hurricane intensity predictability, Bull. Amer. Meteo. Soc., 97 (2016), 1847-1857.  doi: 10.1175/BAMS-D-15-00168.1.  Google Scholar

[13]

C. Kieu, H. Chen and D. L. Zhang, An examination of the pressure-wind relationship for intense tropical cyclones, Wea. and Forecasting, 25 (2010), 895-907.   Google Scholar

[14]

Y. Liu, D.-L. Zhang and M. K. Yau, A Multiscale Numerical Study of Hurricane Andrew Part Ⅱ: Kinematics and Inner-Core Structures, J. Atmos. Sci, 127 (1999), 2597-2616.   Google Scholar

[15]

T. Ma, Topology of Manifolds Science Press, Beijing. , 2007. Google Scholar

[16]

J. W. Milnor, Topology from the Differentiable Viewpoint Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va. 1965.  Google Scholar

[17]

Y. Ogura and N. A. Phillips, Scale analysis of deep and shallow convection in the atmophere, J. Atmos. Sci, 19 (1962), 173-179.   Google Scholar

[18]

R. Rotunno and K. A. Emanuel, An airsea interaction theory for tropical cyclones. part ii: Evolutionary study using a nonhydrostatic axisymmetric numerical model, J. Atmos. Sci, 44 (1987), 542-561.   Google Scholar

[19]

W. Shen, R. E. Tuleya and I. Ginis, A sensitivity study of the thermodynamic environment on GFDL model hurricane intensity: Implications for global warming, J. Climate, 13 (2000), 109-121.   Google Scholar

[20]

R. K. Smith, G. Kilroy and M. T. Montgomery, Why do model tropical cyclones intensify more rapidly at low latitudes, Journal of the Atmospheric Sciences, 72 (2015), 1783-1840.  doi: 10.1175/JAS-D-14-0044.1.  Google Scholar

[21]

R. Wilhelmson and Y. Ogura, The pressure perturbation and the numerical modeling of a cloud, Journal of the Atmospheric Sciences, 29 (1972), 1295-1307.  doi: 10.1175/1520-0469(1972)029<1295:TPPATN>2.0.CO;2.  Google Scholar

[22]

H. E. Willoughby, Forced secondary circulations in hurricanes, J. Geophys. Res, 84 (1979), 3173-3183.  doi: 10.1029/JC084iC06p03173.  Google Scholar

[23]

H. E. Willoughby, Gradient Balance in Tropical Cyclones, J. Atmos. Sci., 47 (1990), 265-274.  doi: 10.1175/1520-0469(1990)047<0265:GBITC>2.0.CO;2.  Google Scholar

Figure 1.  A bifurcation diagram in terms of the $v$ component of the critical points as a function of the stratification parameter $s$ with fixed values $f =0.01$ and $r = 0.01$
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Figure 2.  A bifurcation diagram in terms of the $v$ component of the critical points as a function of the Coriolis parameter $f$ with fixed values $s = 0.1, r = 0.01$
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Figure 3.  a) Flow trajectories for four different initial points in the phase space of $(u, v, b)$ that represent an incipient weak anticyclonic vortex (-0.1, -0.1, 0.1) (red); a mature TC near the MPI equilibrium with a weak warm core (-1, 1, 0.5) (cyan); a mature TC with intensity significant above the MPI equilibrium limit (-1, 1.4, 1) (green); and a mature TC near the MPI equilibrium limit with too weak low level convergence (-0.1, 1, 1) (blue) for the case of $f = 0.05$; (b) Time series of $v$ during the entire simulation; and (c)-(d) Similar to (a)-(b) but for the case of $f=0$
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